On an inequality of Erdős and Turán concerning uniform distribution modulo one. II
type: Item
instance of: scholarly article
title: On an inequality of Erdős and Turán concerning uniform distribution modulo one. II
zbMATH Open document ID: 0813.11044
published in: Journal of Number Theory
publication date: 1994-11-20T00:00:00Z
review text: [Part I, cf. Colloq. Math. Soc. Janos Bolyai 60, 621-630 (1992; Zbl 0791.60013).] Let \(x_ 1, \dots, x_ N\) be a sequence of real numbers; \(\Delta_ N\) its discrepancy and \(\alpha_ k = {1 \over N} \sum^ N_{j=1} e^{2 \pi ik}\) its Fourier coefficients. A famous theorem of Erdős and Turán says \[ \Delta_ N \ll B_ N : = \min_ K \left( {1 \over k} + \sum^{K-1}_{k=1} {| \alpha_ k | \over k} \right). \] The author proves the converse inequality \(\Delta_ N \gg B_ N^{3/2}\) and shows that this bound is best possible by constructing a point set satisfying \(\Delta_ N \ll B_ N^{3/2}\). The construction is based on the Rudin-Shapiro polynomials.
Mathematics Subject Classification ID: 11K06
Mathematics Subject Classification ID: 60E15
zbMATH DE Number: 653309
zbMATH Keywords: Erdős-Turán inequality
zbMATH Keywords: discrepancy
zbMATH Keywords: Rudin-Shapiro polynomials
author: Imre Z. Ruzsa
reviewed by: Robert F. Tichy
MaRDI profile type: Publication
full work available at URL: jnth.1994.1082
OpenAlex ID: W1968903613
DOI: 10.1006/JNTH.1994.1082
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description: scientific article; zbMATH DE number 653309